© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11a158 K11a160

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,4,11,3 X18,6,19,5 X14,8,15,7 X2,10,3,9 X22,11,1,12 X8,14,9,13 X20,15,21,16 X6,18,7,17 X16,19,17,20 X12,21,13,22

Gauss Code:

{1, -5, 2, -1, 3, -9, 4, -7, 5, -2, 6, -11, 7, -4, 8, -10, 9, -3, 10, -8, 11, -6}

DT (Dowker-Thistlethwaite) Code:

4 10 18 14 2 22 8 20 6 16 12

Alexander Polynomial:

t-3 - 9t-2 + 27t-1 - 37 + 27t - 9t2 + t3

Conway Polynomial:

1 - 3z4 + z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{111, 2}

Jones Polynomial:

q-3 - 3q-2 + 7q-1 - 11 + 15q - 18q2 + 18q3 - 15q4 + 12q5 - 7q6 + 3q7 - q8

Other knots (up to mirrors) with the same Jones Polynomial:

{K11a347, ...}

A2 (sl(3)) Invariant:

q-10 - q-6 + 3q-4 - 2q-2 + 3q2 - 4q4 + 2q6 - 2q8 + q10 + 2q12 - 2q14 + 4q16 - q18 - q20 + 2q22 - q24 - q26

HOMFLY-PT Polynomial:

- a-8 + 2a-6 + 3a-6z2 - a-4 - 4a-4z2 - 3a-4z4 + a-2 + 3a-2z2 + 2a-2z4 + a-2z6 - 1 - 3z2 - 2z4 + a2 + a2z2

Kauffman Polynomial:

a-9z - 2a-9z3 + a-9z5 - a-8 + 3a-8z2 - 5a-8z4 + 3a-8z6 + 3a-7z3 - 7a-7z5 + 5a-7z7 - 2a-6 + 5a-6z2 - 4a-6z4 - 3a-6z6 + 5a-6z8 - a-5z + 9a-5z3 - 14a-5z5 + 5a-5z7 + 3a-5z9 - a-4 + 3a-4z2 - 11a-4z6 + 8a-4z8 + a-4z10 + a-3z + 4a-3z3 - 8a-3z5 - 3a-3z7 + 6a-3z9 - a-2 + 3a-2z2 + a-2z4 - 13a-2z6 + 7a-2z8 + a-2z10 + 6a-1z3 - 10a-1z5 + 3a-1z9 - 1 + 5z2 - z4 - 7z6 + 4z8 - az + 6az3 - 8az5 + 3az7 - a2 + 3a2z2 - 3a2z4 + a2z6

V2 and V3, the type 2 and 3 Vassiliev invariants:

{0, 3}

Khovanov Homology: (The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=2 is the signature of 11159. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 17                       1 j = 15                     2   j = 13                   5 1   j = 11                 7 2     j = 9               8 5       j = 7             10 7         j = 5           8 8           j = 3         7 10             j = 1       5 9               j = -1     2 6                 j = -3   1 5                   j = -5   2                     j = -7 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Show[DrawMorseLink[Knot[11, Alternating, 159]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 159]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[18, 6, 19, 5], X[14, 8, 15, 7], > X[2, 10, 3, 9], X[22, 11, 1, 12], X[8, 14, 9, 13], X[20, 15, 21, 16], > X[6, 18, 7, 17], X[16, 19, 17, 20], X[12, 21, 13, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 159]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -9, 4, -7, 5, -2, 6, -11, 7, -4, 8, -10, 9, -3, 10, > -8, 11, -6]
In[5]:=	DTCode[Knot[11, Alternating, 159]]
Out[5]=	DTCode[4, 10, 18, 14, 2, 22, 8, 20, 6, 16, 12]
In[6]:=	alex = Alexander[Knot[11, Alternating, 159]][t]
Out[6]=	-3 9 27 2 3 -37 + t - -- + -- + 27 t - 9 t + t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 159]][z]
Out[7]=	4 6 1 - 3 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 159]}
In[9]:=	{KnotDet[Knot[11, Alternating, 159]], KnotSignature[Knot[11, Alternating, 159]]}
Out[9]=	{111, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 159]][q]
Out[10]=	-3 3 7 2 3 4 5 6 7 8 -11 + q - -- + - + 15 q - 18 q + 18 q - 15 q + 12 q - 7 q + 3 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 159], Knot[11, Alternating, 347]}
In[12]:=	A2Invariant[Knot[11, Alternating, 159]][q]
Out[12]=	-10 -6 3 2 2 4 6 8 10 12 14 q - q + -- - -- + 3 q - 4 q + 2 q - 2 q + q + 2 q - 2 q + 4 2 q q 16 18 20 22 24 26 > 4 q - q - q + 2 q - q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 159]][a, z]
Out[13]=	2 2 2 -8 2 -4 -2 2 2 3 z 4 z 3 z 2 2 4 -1 - a + -- - a + a + a - 3 z + ---- - ---- + ---- + a z - 2 z - 6 6 4 2 a a a a 4 4 6 3 z 2 z z > ---- + ---- + -- 4 2 2 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 159]][a, z]
Out[14]=	2 2 -8 2 -4 -2 2 z z z 2 3 z 5 z -1 - a - -- - a - a - a + -- - -- + -- - a z + 5 z + ---- + ---- + 6 9 5 3 8 6 a a a a a a 2 2 3 3 3 3 3 3 z 3 z 2 2 2 z 3 z 9 z 4 z 6 z 3 4 > ---- + ---- + 3 a z - ---- + ---- + ---- + ---- + ---- + 6 a z - z - 4 2 9 7 5 3 a a a a a a a 4 4 4 5 5 5 5 5 5 z 4 z z 2 4 z 7 z 14 z 8 z 10 z 5 > ---- - ---- + -- - 3 a z + -- - ---- - ----- - ---- - ----- - 8 a z - 8 6 2 9 7 5 3 a a a a a a a a 6 6 6 6 7 7 7 6 3 z 3 z 11 z 13 z 2 6 5 z 5 z 3 z 7 > 7 z + ---- - ---- - ----- - ----- + a z + ---- + ---- - ---- + 3 a z + 8 6 4 2 7 5 3 a a a a a a a 8 8 8 9 9 9 10 10 8 5 z 8 z 7 z 3 z 6 z 3 z z z > 4 z + ---- + ---- + ---- + ---- + ---- + ---- + --- + --- 6 4 2 5 3 a 4 2 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 159]], Vassiliev[3][Knot[11, Alternating, 159]]}
Out[15]=	{0, 3}
In[16]:=	Kh[Knot[11, Alternating, 159]][q, t]
Out[16]=	3 1 2 1 5 2 6 5 q 3 9 q + 7 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 10 q t + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 > 8 q t + 8 q t + 10 q t + 7 q t + 8 q t + 5 q t + 7 q t + 11 5 13 5 13 6 15 6 17 7 > 2 q t + 5 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a159

K11a158 K11a160